Harvey P. Dale - cp's OEIS frontend

A381218 Odd numbers with both an odd number of prime factors (counted with multiplicity) and an odd number of distinct prime factors. (Intersection of A067019 and A098903.)

3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 231, 233, 239

Offset: 1

Harvey P. Dale, Feb 17 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A067019, A098903

Select[Range[1,301,2],AllTrue[{PrimeOmega[#],PrimeNu[#]},OddQ]&]

A381004 Primes ending in 777.

Original entry on oeis.org

1777, 2777, 11777, 19777, 22777, 26777, 41777, 43777, 44777, 47777, 50777, 53777, 65777, 67777, 68777, 71777, 76777, 79777, 80777, 83777, 94777, 97777, 107777, 110777, 113777, 115777, 122777, 124777, 125777, 131777, 134777, 136777, 137777, 145777, 146777

Offset: 1

Harvey P. Dale, Feb 11 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A193552.

select(isprime, [i*1000+777$i=1..150])[];  # Alois P. Heinz, Feb 12 2025

Select[Table[1000 n + 777, {n, 200}], PrimeQ]

select(x->((x % 1000)==777), primes(20000)) \\ Michel Marcus, Feb 12 2025

select(isprime, vector(200, n, 1000*n+777)) \\ Michel Marcus, Feb 12 2025

from itertools import count, islice
from sympy import isprime
def A381004_gen(): # generator of terms
    return filter(isprime,count(777,1000))
A381004_list = list(islice(A381004_gen(),20)) # Chai Wah Wu, Feb 12 2025

a(n) = 1000*A102343(n)+777. - R. J. Mathar, Feb 13 2025

A379371 Primes congruent to 67 mod 12345.

Original entry on oeis.org

67, 123517, 148207, 271657, 296347, 321037, 345727, 395107, 444487, 567937, 666697, 1061737, 1209877, 1407397, 1506157, 1530847, 1827127, 1876507, 1950577, 1999957, 2024647, 2098717, 2222167, 2296237, 2345617, 2469067, 2567827, 2617207, 2691277, 2814727

Offset: 1

Harvey P. Dale, Dec 21 2024

Primes congruent to 67 mod 24690. - Chai Wah Wu, Apr 30 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A301619.

select(isprime, [67+24690*i$i=0..114])[];  # Alois P. Heinz, Dec 21 2024

Mathematica
```
Select[Range[67,3000000,12345],PrimeQ]
```

A378493 Dot product of the first n primes and the first n triangular numbers.

Original entry on oeis.org

2, 11, 41, 111, 276, 549, 1025, 1709, 2744, 4339, 6385, 9271, 13002, 17517, 23157, 30365, 39392, 49823, 62553, 77463, 94326, 114313, 137221, 163921, 195446, 230897, 269831, 313273, 360688, 413233, 476225, 545393, 622250, 704955, 798825, 899391, 1009762

Offset: 1

Harvey P. Dale, Nov 28 2024

nonn

			a(3) = dot product of {2,3,5} and {1,3,6} = 2*1+3*3+5*6 = 41.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Partial sums of A196421.

Cf. A000040, A000217, A166486.

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+ithprime(n)*n*(n+1)/2)
    end:
seq(a(n), n=1..37);  # Alois P. Heinz, Nov 28 2024

Table[Prime[Range[n]].Accumulate[Range[n]],{n,50}]

a(n) mod 2 = A166486(n-1).

A376334 Integer part of the product of three consecutive primes divided by their sum.

Original entry on oeis.org

3, 7, 16, 32, 59, 85, 125, 178, 249, 342, 431, 539, 632, 749, 924, 1102, 1289, 1458, 1645, 1836, 2036, 2324, 2663, 3038, 3352, 3579, 3765, 4005, 4482, 5067, 5770, 6129, 6676, 7123, 7729, 8204, 8775, 9362, 9964, 10515, 11230, 11809, 12499, 12845, 13627, 14792

Offset: 1

Harvey P. Dale, Sep 20 2024

nonn

			a(5) = Floor[(prime(5)*prime(6)*prime(7))/(prime(5)+prime(6)+prime(7))]==59.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A160830.

Floor[Times@@#/Total[#]&/@Partition[Prime[Range[100]],3,1]]

A373799 Index of n-th prime in A374965.

Original entry on oeis.org

2, 5, 9, 14, 19, 22, 25, 36, 38, 43, 47, 51, 56, 65, 72, 74, 76, 97, 100, 102, 105, 107, 110, 112, 115, 122, 125, 128, 130, 238, 255, 260, 272, 284, 286, 290, 293, 296, 300, 316, 325, 331, 562, 565, 567, 575, 578, 607, 610, 612, 617, 627, 632, 649, 651, 654, 866, 875, 878

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Jul 28 2024

nonn

			The fifth prime in order of appearance in A374965 is A375028(5) = 751 = A374965(19), so a(5) = 19.

N. J. A. Sloane, Table of n, a(n) for n = 1..290 (Terms 1 through 289 were obtained using _Harvey P. Dale_'s MMA program for A373798. For a(290), see A375028.)
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A373798 (first differences), A374965, A375028.

from itertools import count, islice
from sympy import isprime, nextprime
def A373799_gen(): # generator of terms
    a, p = 1, 3
    for i in count(1):
        if isprime(a):
            yield i
            a = p-1
        else:
            a = (a<<1)+1
        p = nextprime(p)
A373799_list = list(islice(A373799_gen(),20)) # Chai Wah Wu, Jul 29 2024

A373798 Divide A374965 into "blocks" by saying that each prime term ends a block; sequence gives lengths of successive blocks.

Original entry on oeis.org

2, 3, 4, 5, 5, 3, 3, 11, 2, 5, 4, 4, 5, 9, 7, 2, 2, 21, 3, 2, 3, 2, 3, 2, 3, 7, 3, 3, 2, 108, 17, 5, 12, 12, 2, 4, 3, 3, 4, 16, 9, 6, 231, 3, 2, 8, 3, 29, 3, 2, 5, 10, 5, 17, 2, 3, 212, 9, 3, 4, 5, 22, 3, 5, 13, 5, 9, 4, 12, 8, 2, 57, 2, 65, 5, 3, 93, 9, 46

Offset: 1

Harvey P. Dale and N. J. A. Sloane, Jul 28 2024

nonn

The first 286 terms of the sequence are the result of dividing the first 10000 terms of A374965 into "blocks."
Comment from N. J. A. Sloane, Aug 09 2024 (Start):
Suppose p = A374965(t) is a prime in A374965, and is the s-th prime to appear there (that is, A375028(s) = p and A373799(s) = t). The next term in A374965 is by definition A374965(t+1) = prime(t+1) - 1 = r (say). Then the block starting with r has length a(s+1) = A050412(r) + 1. For example, p = 19 = A374965(5) is the second prime in A374695, so we have s = 2, t = 5, and r = prime(6) - 1 = 13 - 1 = 12. Then A050412(12) = 3, which tells us that a(3) = 3 + 1 = 4. The block is [12, 25, 51, 103].
For a larger example, the s = 285th prime in A374965 is p = 160077823 = A374965(7686), so t = 7686. The next block begins with r = prime(7687) - 1 = 78282. After 39 steps of double-and-add-1 (corresponding to A050412(78282) = 39) we reach the 286th prime in A374965, A374965(7726) = 43036534378594303. (End)

			A374965 begins
1, 3/ 4, 9, 19/ 12, 25, 51, 103/ 28, 57, 115, 231, 463/ 46, 93, 187, 375, 751/ 70, 141, 283/ 82, 165, ...,
where the primes are followed by slashes, to indicate the blocks. The lengths of the initial blocks are 2, 3, 4, 5, 5, 3, ...

N. J. A. Sloane, Table of n, a(n) for n = 1..289 [Terms 1 to 286 from Harvey P. Dale] Obtained using Harvey P. Dale's MMA program.

Cf. A374965, A375028.

nxt[{n_, a_}] := {n + 1, If[! PrimeQ[a], 2 a + 1, Prime[n + 1] - 1]}; vvv=NestList[nxt,{1,1},9999][[;;,2]]; Total/@Partition[Length/@SplitBy[vvv,PrimeQ],2] (* Harvey P. Dale, Jul 28 2024 *)

A375028 Primes in A374965 in order of their occurrence.

Original entry on oeis.org

3, 19, 103, 463, 751, 283, 331, 103423, 313, 2671, 1543, 1783, 3823, 68863, 20287, 733, 757, 407896063, 2083, 1093, 2251, 1153, 2371, 1213, 2467, 41023, 2707, 2803, 1453, 119909605576788675546376149602926591, 98238463, 25903, 3405823, 3590143, 3733, 14983, 7603, 7723, 15607, 65306623, 537343, 69151, 3859801644442622798122887215978426484283282692686288680974641672159756287

Offset: 1

Harvey P. Dale and N. J. A. Sloane_, Jul 28 2024

nonn

Sequences A050412 and A052333 suggest that it is possible that the present sequence has only finitely many terms. - N. J. A. Sloane, Jul 29 2024

Harvey P. Dale, Table of n, a(n) for n = 1..289 [This b-file ends with a(289) = 838951. Thanks to the work of _Lucas A. Brown_ (see A050412), we can now say that the next term a(290) is the 102410-digit prime 104917*2^340181 - 1. Of course this is too large to include in a b-file. - _N. J. A. Sloane_, Jul 31 2024]
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A374965; A050412, A052333.
A373799 gives the indices where the primes appear in A374965.
A373804 gives the primes sorted into increasing or5der.

nxt[{n_,a_}]:={n+1,If[!PrimeQ[a],2a+1,Prime[n+1]-1]}; Select[NestList[nxt, {1, 1}, 999][[;; , 2]], PrimeQ]

from itertools import islice
from sympy import isprime, nextprime
def A375028_gen(): # generator of terms
    a, p = 1, 3
    while True:
        if isprime(a):
            yield a
            a = p-1
        else:
            a = (a<<1)+1
        p = nextprime(p)
A375028_list = list(islice(A375028_gen(),30)) # Chai Wah Wu, Jul 29 2024

A370867 Records in A224908.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 11, 12, 14, 15, 19, 20, 24, 27, 32, 39, 44, 49, 50, 52, 56, 57, 66, 68, 87, 103, 109, 121, 127, 148, 152, 163, 197, 216, 235, 260, 283, 325, 356, 374, 395, 406, 437, 485, 488, 494, 566, 569, 617, 644, 658, 673, 747, 770, 774, 809, 820, 826

Offset: 1

Harvey P. Dale, Mar 03 2024

nonn

Cf. A224908.

DeleteDuplicates[Table[Count[Prime[n]+Prime[Range[n-1]]+1,_?PrimeQ],{n,3000}],GreaterEqual]

A370139 Primes p such that the sums of three, five, and seven consecutive primes starting with p are prime.

Original entry on oeis.org

19, 29, 31, 53, 79, 379, 401, 839, 883, 1301, 1409, 1951, 1973, 2113, 2683, 2791, 2833, 3407, 3613, 3793, 3823, 4441, 4751, 4831, 5623, 5827, 6133, 6329, 7187, 7237, 7703, 8527, 9173, 10103, 10853, 11317, 12277, 13163, 13933, 14159, 14827, 15241, 15667

Offset: 1

Harvey P. Dale, Feb 11 2024

nonn

			379 is in the sequence because the seven consecutive primes starting with 379 are 379, 383, 389, 397, 401, 409, and 419, and (379+383+389)=1151, and (379+383+389+397+401)=1949, and (379+383+389+397+401+409+419)=2777, and 1151 and 1949 and 2777 are all primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A180948 and A182121.

Select[Partition[Prime[Range[5000]],7,1],AllTrue[{Total[Take[#,3]],Total[Take[#,5]],Total[#]},PrimeQ]&][[;;,1]]

cp's OEIS Frontend

User: Harvey P. Dale

A381218 Odd numbers with both an odd number of prime factors (counted with multiplicity) and an odd number of distinct prime factors. (Intersection of A067019 and A098903.)

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A381004 Primes ending in 777.

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A379371 Primes congruent to 67 mod 12345.

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Maple

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A378493 Dot product of the first n primes and the first n triangular numbers.

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Maple

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Formula

A376334 Integer part of the product of three consecutive primes divided by their sum.

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Mathematica

A373799 Index of n-th prime in A374965.

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Python

A373798 Divide A374965 into "blocks" by saying that each prime term ends a block; sequence gives lengths of successive blocks.

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Mathematica

A375028 Primes in A374965 in order of their occurrence.

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